Get a dog! Or get a tape of a dog barking. Do they have kids? Do you? Do you belong to the same shul? The same yeshiva? You might ask the Rav to talk to them.
As far as the landlord is concerned it doesn’t seem as if he cares much about you nor does it seem like he cares if you move out. Ask your neighbors if they know of any other apartments in the area. If they ask you why you are moving simply say the neighbors upstairs are not very considerate about running their machinery late at night and early in the morning. You just can’t handle the noise any more.